Two-step synthesis of N-sulfonyl aziridines from epoxides

Article abstract of DOI:10.1016/j.jqsrt.2006.02.064

A convenient and high-yielding two-step synthesis of N-sulfonyl aziridines starting from epoxides is described. The method, which involves epoxide ring opening with sulfonamides and subsequent mesylation-cyclisation, is particularly suitable for variation

Full text of DOI:10.1016/j.jqsrt.2006.02.064

ARTICLE IN PRESS
Journal of Quantitative Spectroscopy &
Radiative Transfer 102 (2006) 425–431
www.elsevier.com/locate/jqsrt
The output of amplified spontaneous emission lasers
Ã
H. Huang, G.J. Tallents
Department of Physics, University of York, York, YO10 5DD, UK
Received 8 September 2005; accepted 6 February 2006
Abstract
The output of amplified spontaneous emission (ASE) lasers such as X-ray lasers operated without mirrors is calculated
exactly for Gaussian and Lorentzian small signal gain profiles by a simple Taylor series expansion. The accuracy of the
‘Linford’ formula commonly used as an approximation for the output of ASE lasers is evaluated by comparison to our
exact solutions. The Linford formula is accurate to better than 10% for intensities produced by a Gaussian gain profile,
pffiffiffi
but requires multiplication by a correction factor of 1= p; at gain length product greater than 5 for Lorentzian gain
profiles.
r 2006 Elsevier Ltd. All rights reserved.
Keywords: X-ray; Laser; ASE; Plasma
1. Introduction
The propagation of X-ray laser beams has been investigated in a number of papers [1]. X-ray lasers usually
operate without mirrors and the propagation of the beam is determined by the theory of amplified
spontaneous emission (ASE) [2]. The small signal gain coefficient g0 at line centre of an ASE medium, specifies
the amplification of intensity achieved. Early studies of X-ray lasers strove to achieve a product of the small
signal gain with the length L of the laser medium such that g0L was at a value where saturation occurred.
Saturation of a laser occurs when stimulated emission depletes the population of the upper quantum state and
typically occurs for g0LE15 [3].
X-ray lasers have narrow spectral profiles (u/DuX104) that cannot be readily spectrally resolved.
Nevertheless, Koch et al. [4] constructed a grating spectrometer with resolving power u/DuE35,000 and
measured X-ray laser spectral line widths. Other X-ray laser spectral profiles have been resolved using the
variation of fringe visibility in interferometry [5,6]. However, most experiments record intensities integrated
over the spectral profile of the X-ray laser output.
In the evaluation of experiments, refraction due to spatially varying refractive indices often needs to be
considered in the X-ray laser propagation process. Free electrons usually dominate the refractive index and so
the refractive index for X-ray lasers is usually close to unity (although an exception has been recently observed
[7]). However, due to the long propagation lengths (ꢀ1–20 mm) required to achieve high X-ray laser output,
Ã
Corresponding author. Tel.: +44 1904 432286; fax: +44 1904 432214.
E-mail address: Gjt5@york.ac.uk (G.J. Tallents).
0022-4073/$ - see front matter r 2006 Elsevier Ltd. All rights reserved.
doi:10.1016/j.jqsrt.2006.02.064

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plasma electron density gradients can refract an X-ray laser beam out of the gain region if the gradients are
not reduced by using a pre-pulse to pre-form a plasma with low gradients [8]. If the plasma width approaches
the wavelength of the laser, a wave optic treatment of the X-ray laser propagation may be required [9].
However, there are a range of situations in X-ray laser media where a simple ray optic treatment [2,10–12]
neglecting refraction is adequate.
In this paper, we consider the X-ray laser medium as comprising a column of length L of uniform gain so
that a single ray treatment is appropriate. Linford [10] evaluated an approximate treatment for this situation
giving the output intensity as a function of the small signal gain after integrating over the spectral output, but
neglecting saturation effects. The formula determined by Linford [10] has been used in more than 100 papers
to evaluate the output of X-ray lasers (for example [9,13–15]), but has not been compared to exact solutions in
the published literature. Casperson [11] and Pert [2] determined formulae to evaluate the total output intensity
of ASE lasers including saturation. The developed formulae relate the peak gain coefficient and the small
signal gain coefficient. In the short gain duration situation, the pulse cannot propagate along the gain length
during the gain duration [1], so Strati and Tallents [12] evaluated time dependent effects on X-ray laser output
associated with a short gain duration.
2. The radiative transfer equations
This paper assumes that a small signal gain coefficient is constant over a cross-section area and along a
length L of a plasma column. The X-ray laser propagation can be approximated as occurring in one direction
(the z direction), along the plasma length [1]. In a one-dimensional treatment, we calculate the forward
irradiance If(u) in a positive z-direction and the backward irradiance Ib(u) at frequency u in the negative z-
direction by the following [1]:
dIf ðuÞ
dz
¼ GðuÞIf ðuÞ þ EðuÞ,
dIbðuÞ
dz
¼ GðuÞIbðuÞ þ EðuÞ,
ð1Þ
where G(u) is the gain coefficient and E(u) the spontaneous emission per unit length.
The gain coefficient G(u) is related to the small signal gain g0 at the line centre by
g0
f ðuÞ
,
GðuÞ ¼
(2)
1 þ ðIav=IsÞ f ð0Þ
where Iav is the X-ray laser irradiance averaged over the line profile, Is the saturation intensity, and Iav is given
by
Z
f ðuÞ
Iav
¼
IðuÞ du,
(3)
f ð0Þ
where f(u) is the line profile variation with frequency u. We consider Lorentzian and Gaussian line profile.
Assuming frequency u ¼ 0 corresponds to the line centre, a Lorentzian profile [16] is written as
DuL
2p
1
f LðuÞ ¼
,
(4)
2
u2 þ ðDuL=2Þ
where DuL is the full-width at half-maximum of the Lorentzian profile. At line centre
2
f Lð0Þ ¼
.
pDuL
A Gaussian profile [17] is written as
pffiffiffiffiffiffiffiffiffiffiffiffi
ꢀ
ꢁ
4 ln 2
u2
pffiffiffi
f GðuÞ ¼
exp ꢁ4 ln 2
,
(5)
2
DuG
p
ðDuGÞ

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427
where DuG is the full-width at half-maximum of the Gaussian profile. At line centre
pffiffiffiffiffiffiffiffiffiffiffiffi
4 ln 2
pffiffiffi
solution for the total X-ray laser output by integrating Eq. (1) over frequency and over the length L We have
f Gð0Þ ¼
.
DuG
p
R
Both expressions (Eqs. (4) and (5)) for line profiles are normalized so that þ1 f ðuÞ ¼ 1. We can achieve a
ꢁ1
Z
Z
È
É
Itot
¼
If ðuÞ du ¼ I0f ð0Þ
¼ I0f ð0Þ
exp½GðuÞLꢂ ꢁ 1 du
þ1ꢄ
ꢂ
ꢃ
ꢅ
Z
f ðuÞ
f ð0Þ
exp Gð0ÞL
ꢁ 1 du,
ð6Þ
ꢁ1
where I0f ð0Þ ¼ EðuÞ=GðuÞ.
Using the Taylor expansion of the exponential term in Eq. (6) and assuming a Gaussian gain profile
f ðuÞ ¼ f GðuÞ, we can evaluate Eq. (6)
þ1ꢂ
ꢃ
Z
2
GðuÞL ½GðuÞLꢂ
½GðuÞLꢂ
n!
Itot ¼ I0f Gð0Þ
þ
þ ꢃ ꢃ ꢃ þ
ꢂ
n þ ꢃ ꢃ ꢃ du
1!
2!
ꢁ1
(
)
ꢀ
ꢁꢃ
Z
2
¼ I0f ð0Þ þ1
exp ꢁ4ln 2
n
du
n
þ1
X
½Gð0ÞLꢂ
u
DuG
G
n!
ꢁ1
n¼1
pffiffiffi
þ1
n
X
DuG
p
½Gð0ÞLꢂ
pffiffiffiffiffiffiffiffiffiffiffiffi
pffiffi
¼ I0f Gð0Þ
nn!
4 ln 2 n¼1
þ1
n
X
½Gð0ÞLꢂ
pffiffiffi
¼ I0
.
ð7Þ
nn!
n¼1
Each term of Eq. (7) for a particular G(0)L can be evaluated (see Fig. 1). From plots such as Fig. 1, we can
see that the Taylor series expansion for the Gaussian profile is convergent. To achieve a sufficiently accurate
total intensity evaluation, we need to sum up to 40 terms in n.
Pert [2] and Casperson [11] developed an approximation of the small signal gain length product including
the effect of saturation on X-ray laser output to obtain an expression relating the actual gain length product
G(0)L=11
8.0x104
G(0)L=13
G(0)L=15
6.0x104
4.0x104
2.0x104
0.0
0
20
40
60
80
100
n
Fig. 1. The value of each term of the calculated intensity using equation (7) assuming a Gaussian gain profile for particular Gð0ÞL ¼ 11, 13
and 15.

ARTICLE IN PRESS
H. Huang, G.J. Tallents / Journal of Quantitative Spectroscopy & Radiative Transfer 102 (2006) 425–431
428
106
105
104
103
102
101
100
0
5
10
15
20
25
30
g0L
Fig. 2. Irradiance output as a function of small signal gain length product for a Gaussian profile gain coefficient and I0=Is ¼ 10ꢁ6
.
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0
2
4
6
8
10
12
14
16
18
20
22
24
G (0) L
Fig. 3. The value of g needed to correct the Linford formula for a Gaussian gain profile as a function of gain length product (G(0)L). The
value of g ¼ 1 asymptotically correct for small and large G(0)L is also shown.
G(0)L to the small signal gain length product g0L. We can write
ꢀ
ꢁ
I0
Is
Itot
Is
1 ꢁ 2
Gð0ÞL þ 2
¼ g0L,
(8)
where I0=IS ffi O=ð4pÞ [1] for O representing the solid angle into which spontaneous emission can be amplified
[1]. Typically, I0/IsE10ꢁ6. Eq. (8) enables us to evaluate Itot as a function of the small signal gain length I0L
using Eq. (7) (see Fig. 2). We can see from Fig. 2 that saturation occurs at Gð0ÞL ꢄ g0l ¼ 15218.
Linford et al. [10] calculated an approximation to Eq. (6), commonly known as the Linford approximation
(see for example [13]). The approximation is strictly only applicable to a Gaussian profile. It was obtained by
matching asymptotic G(0)L-0 and G(0)L-+N expressions to give
3=2
fexp½Gð0ÞLꢂ ꢁ 1g
Itot ¼ I0g
.
(9)
1=2
fGð0ÞL exp½Gð0ÞLꢂg
A factor gE1 can be introduced to allow for any deviation of Itot calculated by integrating Eq. (7) and the
value predicted by the Lindford formula (Eq. (9)). The exact value of g for a Gaussian profile gain coefficient is

ARTICLE IN PRESS
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429
shown in Fig. 3. It can be seen that g is within 10% of unity for a Gaussian gain profile and hence that the
Linford formula is a good approximation.
3. Radiative transfer with a Lorentzian gain profile
For a Lorentzian gain profile, Eq. (6) becomes
þ1ꢄ
ꢂ
ꢃ
ꢅ
ꢁ 1 du
#
Z
f LðuÞ
f Lð0Þ
Itot ¼ I0f Lð0Þ
¼ I0f Lð0Þ
exp Gð0ÞL
ꢁ1
þ1
(
"
)
Z
1
exp Gð0ÞL
ꢁ 1 du.
ð10Þ
2
1 þ ð2u=DuLÞ
ꢁ1
Let
f LðuÞ
1
¼
¼ cos2y,
(11)
2
f Lð0Þ
1 þ ð2u=DuLÞ
then du ¼ ðDuL=2Þðcosꢁ2yÞ dy and we have
ꢂ
ꢃ
Z
Itot ¼ 2I0f Lð0Þ p=2fexp½Gð0ÞL cos2yꢂ ꢁ 1g
ðcosꢁ2yÞ dy
DuL
2
0
ꢄ
Z
p=2
2
½ðGð0ÞLÞ ꢃ cos2yꢂ
¼ DuLI0f Lð0Þ
ðGð0ÞLÞcos2y þ
2!
0
ꢅ
n þ ꢃ ꢃ ꢃ ðcosꢁ2yÞ dy
½Gð0ÞLcos2yꢂ
þ ꢃ ꢃ ꢃ þ
n!
(
)
ꢂ
ꢃ
Z
1
n
p=2
X
ðGð0ÞLÞ
¼ DuLI0f Lð0Þ
cos2ny ðcosꢁ2yÞ dy
n!
0
n¼1
(
)
ꢂ
ꢃ
Z
1
n
p=2
X
Â
Ã
ðGð0ÞLÞ
DuLI0f Lð0Þ
cos2nꢁ2y dy .
ð12Þ
n!
0
n¼1
5x104
4x104
3x104
2x104
1x104
G(0)L=11
G(0)L=13
G(0)L=15
0
0
20
40
60
80
100
n
Fig. 4. The value of each term of the calculated intensity using equation (14) assuming a Lorentzian gain profile for particular Gð0ÞL ¼ 11,
13 and 15.

ARTICLE IN PRESS
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430
R
Â
Ã
p=2
0
Integrating
Z
cos2nꢁ2y dy gives
ꢆ
ꢆ
À
Á
3
p=2
ꢆ
ꢆ
Â
Ã
n ꢁ 2
!
cos2nꢁ2y dy ¼ p
,
(13)
ðn ꢁ 1Þ!
0
so
ꢆ
ꢆ
ꢆ
ꢆ
À
Á
ꢄꢂ
ðGð0ÞLÞ
n!
ꢃ
ꢅ
1
n
3
X
n ꢁ 2
!
ðGð0ÞLÞ
n!
Itot ¼ DuLpI0f Lð0Þ
ðn ꢁ 1Þ!
n¼1
ꢆ
ꢆ
ꢆ
À
Á
ꢄꢂ
ꢃ
ꢅ
1
n
3
ꢆ
X
upon using Eq. (4).
n ꢁ 2
!
¼ 2I0
.
ð14Þ
ðn ꢁ 1Þ!
n¼1
Each term of Eq. (14) for a particular G(0)L can be evaluated (see Fig. 4). We can see that the Taylor series
expansion for the Lorentzian profile is convergent. To achieve a sufficiently accurate total intensity evaluation,
we again need 40 terms in n. We can evaluate Itot as a function of the small signal gain coefficient product (see
Fig. 5).
106
105
104
103
102
101
100
0
5
10
15
g0L
20
25
30
Fig. 5. Irradiance output as a function of small signal gain length product assuming a Lorentzian gain profile and I0=Is ¼ 10ꢁ6
.
1.5
1.4
1.3
1.2
1.1
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0
2
4
6
8
10
12
14
16
18
20
22
24
G (0) L
Fig. 6. The value of g needed to correct the Linford formula for a Lorentzian gain profile as a function of gain length product (G(0)L). The
pffiffiffi
asymptotic value g ¼ 1= p is also shown.

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431
The value of g (see Eq. (9)) needed to modify the Linford formula for a Lorentzian gain profile is shown in
Fig. 6. At small G(0)L the Linford formula is a good approximation, but at large G(0)L the value of g
pffiffiffi
approaches 1= p ꢄ 0:565 [2]. For experiments where G(0)L45, it seems best to use the Linford formula with
pffiffiffi
g ꢄ 1= p for Lorentzian gain profiles. For G(0)Lo5, Fig. 6 could be employed or Eq. (14) evaluated to
deduce g.
4. Conclusion
The accuracy of the Linford formula used to evaluate the output of amplified spontaneous lasers such as X-
ray lasers has been evaluated for Gaussian and Lorentzian gain profiles. The Linford formula is accurate to
better than 10% for intensities produced by a Gaussian gain profile, but requires multiplication by a
pffiffiffi
correction factor of 1= p at gain length product greater than 5 for Lorentzian gain profiles.
Acknowledgement
Funding has been provided by the UK Engineering and Physical Science Research Council.
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